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versión impresa ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009

A description of hereditary skew group algebras of Dynkin and Euclidean type

Olga Funes

Abstract. In this work we study the skew group algebra Λ[G] when G is a finite group acting on Λ whose order is invertible in Λ. Here, we assume that Λ is a finite-dimensional algebra over an algebraically closed field k. The aim is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group algebra for each action and finally to determinate their representation type.

1. Introduction

In this work we assume that is a finite-dimensional algebra over an algebraically closed field . Let a finite group acting on . The skewgroup algebra is the free left -module with basis all the elements in and multiplication given by for all in , in . We study the skew group algebra when is a finite group acting on whose order is invertible in .

There is an extensive literature about skew group algebras and crossed product algebras , and their relationship with the ring , given by elements in that are fixed by . It is of interest to study which properties of are inherited by , or . Some of these ideas are rooted in trying to develop a Galois Theory for non-commutative rings. See [1, 3, 7, 9, 10, 11, 14, 13] for more details.

It is of interest to find ways to describe in terms of because the algebras and have many properties in common which are of interest in the representation theory of finite-dimensional algebras, like finite representation type, being hereditary, being an Auslander algebra, being Nakayama, see [2, 16] for more details. However, we must observe that there are properties which are not preserved by this construction, like being a connected algebra, so we are dealing with essentially different algebras.

It is well known [6] that a connected hereditary algebra is of finite representation type if and only if the underlying graph of its quiver is one of the Dynkin diagrams (), (), , or ; some years later it was shown that a a connected hereditary algebra is of tame representation type if and only if the underlying graph of its quiver is one of the euclidean diagrams (), (), , or , see [4, 12, 17].

The aim of this paper is to describe all possible actions of a finite abelian group on an hereditary algebra of finite or tame representation type, to give a description of the resulting skew group algebra for each action and finally to determinate their representation type.

Then, in order to classify the finite and tame representation type hereditary skew group algebras, it suffices to study the group actions on the Dynkin and the euclidean quivers. In order to do this description, we start by considering a short exact sequence of groups . We can express in terms of the skew group algebra or the crossed product algebra . In this context, we describe when is isomorphic to , for a finite group whose order is invertible in . In section we provide an introduction to the subject, that is, the definition of skew group algebra and crossed product algebra. Finally, in section we consider hereditary algebras of finite representation type and in section we consider hereditary algebras of tame type. In each one of these cases, that is, when the associated quiver is (), (), , , , (), (), , or , we get a connection between and the crossed product algebra with a complete description of all the possible groups appearing in each case, where is the subgroup of consisting on all the elements acting trivially on a complete set of primitive orthogonal idempotents of the algebra . As a consequence of all these results, we get that if acts trivially on then the crossed product algebras obtained in each description are skew group algebras. Finally, the case is considered at the end of section 4.

2. Skew group algebras

This section consists of the preliminaries necessary for the proof of the main results.

Let be a finite-dimensional -algebra and a finite group acting on . The skew group algebra is the free left -module with basis all the elements in and multiplication given by for all in , in . Clearly is a finite-dimensional -algebra. If we identify each in with in and each in with in , we have that is the group of units of and is a -subalgebra of .

Let be a basic finite-dimensional algebra (associative with unity) over an algebraically closed field. A quiver is a quadruple consisting of two sets (whose elements are called points, or vertices ) and (whose elements are called arrows), and two maps which associate to each arrow its source and its target . An arrow of source and target is usually denoted by . A quiver is usually denoted briefly by or even simply by . Thus, a quiver is nothing but an oriented graph without any restriction as to the number of arrows between two points, to the existence of loops or oriented cycles.

We write a path in as a composition of consecutive arrows where for all , and we set , . The path algebra is the -vector space with basis all the paths in , including trivial paths of length zero, one for each vertex . The multiplication of two basis elements is the composition of paths if they are composable, and zero otherwise. A relation from to is a linear combination such that, for each , is a non-zero scalar and a path of length at least two from to . A set of relations on generates an ideal , said to be admissible, in the path algebra of .

It is well-known that if is basic there exists a quiver and a surjective algebra morphism whose kernel is admissible, where , the ordinary quiver of , is defined as follows:

If is a complete set of primitive orthogonal idempotents of , the vertices of are the numbers which are taken to be in bijective correspondence with the idempotents ;

Given two points the arrows are in bijective correspondence with the vectors in a basis of the -vector space .

If is acting on a basic algebra , we can view as in such a way that the action of on is induced by an action of on which leaves stable and preserves the natural grading on by the length of paths. Then is isomorphic to , see [16, Proposition 2.1]. Moreover, if contains no multiple arrows, the action of on is simple: each permutes the vertices in and maps each arrow onto a multiple scalar of the unique arrow from to . From now on we assume with the action of as described above, without double arrows.

Proposition 2.1.Letbe a finite group acting on, letbe theassociated quiver of,without double arrows, and. Weconsider the action ofoninduced by an automorphism of algebraswhich preserves the length of paths of. Then

If,, thenfor some;

Ifand, thenfor some arrow,. In particular, iffixes the starting andending point ofthen, with;

Ifis a source (sink) thenis a source (sink);

The cardinal of the set of arrows that start (end) inis equalto the cardinal of the set of arrows that start (end) in.

Proof.

Let be a primitive idempotent in . Since the action of preserves the vector space generated by arrows, . Moreover, , then we have that , and hence , that is, . On the other hand, suppose . Then

But this is a contradiction because es primitive. Then for some .

If , with . Moreover if then , for some arrow , because has no double arrows. Then .

Let be an idempotent of . Suppose that is a source and is not. Then, there exists an arrow such that , that is, there exists an index such that . Since the action of on is induced by an automorphism of algebras which preserves the length of paths, there exists an arrow such that . Then we have because is a source. This contradiction arises from the assumption that is not a source. Similarly we prove that if is a sink then is a sink.

Let and be the -vector space with basis . If , by ii) we know that the automorphism induces an isomorphism . Then the cardinal of and are equal.

2.1. Crossed product algebra. The purpose of this section is to present the crossed product algebras in order to study when is isomorphic to where is a short exact sequence of groups. We start with the definition of crossed product algebras and prove, for completeness, Theorem 2.2 that connects the skew group algebras with the crossed product algebras. See [16] for more details.

Let be a ring, a finite group acting on , the units of and , a map satisfying

for ,,;

for , the identity element of ;

for , .

Then the corresponding crossed product algebra has elements ; . Addition is componentwise, and multiplication is given by and .

Let be a group and be a short exact sequence of groups. Let be a disjoint union of lateral classes. Then where , .

Theorem 2.2.Ifis a short exact sequenceof groups, then

where the action ofonis induced by the action of, theaction ofonis defined by

andis defined by, with.

Proof. We consider the action of on induced by the action of and the action of on given by . We claim that the action is well defined since is a normal subgroup of . If , then for some , that is . Let be defined by

A direct computation shows that is a crossed product. Now let us see that the map given by

is an isomorphism of -algebras, where . Clearly is a morphism of -vector spaces. If , then

because is a normal subgroup of . On the other hand we have

and agrees with . Furthermore it is clear that is bijective, hence we get .

Corollary 2.3.If is a short exactsequence of groups that splits on the right, then

Proof. We only have to prove that the map defined in the theorem above is such that for any , where , with for some . If the sequence splits on the right, there exists a map such that . Let . Since , then and hence we assume that . Since is a morphism of groups, and therefore . Now it is clear that .

Corollary 2.4.Iforthen

It is clear that the map which defines a crossed product is by definition a cocycle with respect to group cohomology, see [8] for more details. We shall prove that if the cocycle is a coboundary, then we have .

Proof. A direct computation shows that defines a crossed product. Let be defined by . Then is a morphism of algebras because

Therefore is an isomorphism and hence .

We say that acts trivially on an element if for all . If is a finite abelian group of order acting trivially on with invertible in , then . In fact, by Maschke's theorem we have that , see [14]. Now the map given by is an isomorphism of -algebras.

Proposition 2.6. [16, Proposition 5.8] Letbe a finite cyclic groupacting on a commutative local algebrawith the order ofinvertible in. Then.

We may infer from the previous proposition that if is a finite cyclic group with the order of is invertible in , and (the center of ) is a local algebra, because any cocycle is a coboundary. In particular, if is a basic connected algebra without oriented cycles, .

Corollary 2.7.Letbe a finite abelian group acting on a basicconnected algebrawhere the associated quiver has no orientedcycles, with the order ofinvertible in. Letbe a subgroupofwhich acts trivially on, withcyclic. Then.

Proof. It follows from Theorem 2.2 that , and by Maschke's theorem. Since is abelian and acts trivially on , takes values in the set of invertible elements of the center of . But and is cyclic, so Proposition 2.6 implies that is a coboundary. From Proposition 2.5 we may deduce that .

It is known that if is a finite group of order acting trivially on the idempotents of and is invertible in , then is an abelian group, see [15, Proposition 2.7]. In fact, given , with a -root of unity. Hence , and this equality holds for any arrow . Moreover, for every . So .

Finally, we state a result that will be used in the proof of the main theorem in this work.

3. with an abelian group and an hereditary algebra of finite representation type

The aim of this section is to describe all possible actions of a finite abelian group on an hereditary algebra of finite representation type and to give a description of the skew group algebra for each action.

Gabriel has shown in [6] that a connected hereditary algebra is representation-finite if and only if the underlying graph of its quiver is one of the Dynkin diagrams , (), , or , that appear also in Lie theory, where the index in the Dynkin graph always refers to the number of points in the graph. Then, in order to classify the representation-finite hereditary skew group algebras, it suffices to study the group actions on the Dynkin quivers.

Before we present the results, we need some definitions.

Definition 3.1.We say that an quiver of typehas symmetricorientation if it is symmetric with respect to the middle point.

Definition 3.4.We say that the quiverof typehas symmetricorientation if it is symmetric with respect to the side, that is,

and, orand;

and,

, or.

Remark 3.5.

The quiveris symmetric with respect to the middle pointif that point is center of symmetry of the quiver.

The quiveris symmetric with respect to the sideifthe line obtained with the pointsis a symmetry axis ofthe quiver.

As we have already mentioned, if is acting trivially on , we have . Hence, from now on, we will assume that is acting non trivially on . Let . Clearly is a normal subgroup of . Let , then is a short exact sequence of groups.

Ifandwithof typethenis oftype, with symmetric orientation, the order ofiseven and;

Ifandwithof type,, thenhas symmetric orientation, the order ofis even and;

Ifandwithof typethen

has symmetric orientation of kind, the order ofis even and,

or

has symmetric orientation of kind, the order ofis divisible byor, and or

Ifandwithof typethenhassymmetric orientation,is a group of even order and;

Ifwithof typeorof typethenand.

Proof. In order to prove the theorem, we need a precise description of all the possible actions of on , for each type and orientation of . We use Proposition 2.1 to describe all possible actions of on with of type , , , or .

Let with of type and let , . If then . This implies and with -roots of unity. Repeating this procedure we have that implies with an -root of unity, and this for all . Hence the action of is trivial on the idempotents of . So , a contradiction. So . In this case and , will have to be sinks or sources, see Proposition 2.1. This determines the orientation of and . Moreover . So and . Inductively, and , and this for all , with , . If is an even number we have that , and if is the arrow , then , contradiction. We also get a contradiction if . Then if the number of vertices is an even number, the unique possible action on the set of idempotents is the trivial one.

Let with acting non trivially on the set of idempotents of . Then and , hence the quiver has symmetric orientation. Moreover, for all , so . Then has even order and , for all . Let , . Since , does not act trivially on the set of idempotents of . By the previous reasoning, the unique non trivial action is given by . Then . As a consequence , that is . Then because , and hence . Hence, if the group does not act trivially on the set of idempotents of , in accordance with the previous analysis we have that is an even number and is of type with symmetric orientation. In this case we have . Hence , see Theorem 2.2 and Theorem 2.8.

Let with of type , . Assume that the group is not acting trivially on the set of idempotents of . We observe that all must satisfy , see Proposition 2.1. If then , and for all . This determines the orientation of the arrows, that is, has symmetric orientation, and , , for all , with -roots of unity, non zero. Then for all , that is, . So has even order and . Let , . By the previous reasoning, and act in the unique possible non trivial way on the complete set of idempotents of . Then , and for all . Hence , that is , then because . Hence and is an even number.

It follows from the previous analysis that is an even number and the quiver has symmetric orientation. Hence we have .

In fact , and , so in . On the other hand, since for all with and and is abelian, we have that cannot contain simultaneously elements acting as for all with and . Consequently or . The cases i), ii) and iii) determine the orientation of the arrows and , that is, has symmetric orientation of kind (a) or (b), and the cases iv) and v) determine the orientation of all the arrows, that is, has symmetric orientation of kind (b).

In accordance with Definition 3.3 and with the previous analysis for of type , we have that the quiver has symmetric orientation of kind (a) and , or has symmetric orientation of kind (b) and or . From Theorem 2.2 and Theorem 2.8 we have that, in the first case, and . In the second case, the order of is divisible by or , or , and or

We need again a precise description of all the possible actions of on with of type . Let , . By Proposition 2.1 , , and this implies that . On the other hand or . If , then and . This is a contradiction, because . Then , and this implies that and . This determines the orientation of the arrows, and we have , , , and with non zero and an -root of unity. Since for all , then . So has even order and , . Let be such that . Hence, for all . Therefore , and that is, , and then . Hence, if the group does not act trivially on the set of idempotents of , in accordance with the previous analysis, we have that is an even number, has symmetric orientation and . Hence , see Theorem 2.2 and Theorem 2.8.

If we consider the cases of type or , the unique possible action on the set of idempotents is the trivial one. Hence and and the result follows from i).

Letbe a field such that. Ifis anhereditary algebra withof typeandisacting non trivially on the setof idempotents of, then the skew group algebrais Morita equivalentto an algebrawithof type. Ifisacting non trivially on the setof idempotents of, then the skew group algebrais Morita equivalentto an algebrawithof type.

Letbe a field such that. Ifis anhereditary algebra, withof typeandisacting non trivially on the setof idempotentsof, then the skew group algebrais Moritaequivalent to an algebrawithof typeifand of typeif.

Letbe a field such that. Ifisan hereditary algebra, withof type,, andis acting non trivially on the setofidempotents of, then the skew group algebraisMorita equivalent to an algebrawithof type.

Letbe a field such that. Ifis anhereditary algebra, withof type, andisacting non trivially on the setof idempotents of, then the skew group algebrais Morita equivalentto an algebrawithof type.

4. with an abelian group and an hereditary algebra of tame representation type

The aim of this section is to describe all possible actions of a finite abelian group on an hereditary algebra of tame representation type, to give a description of the skew group algebra for each action and finally to determinate their representation type.

It is well known that a connected hereditary algebra is of tame representation type if and only if the underlying graph of its quiver is one of the euclidean diagrams (), (), , or where an euclidean diagram has points. Then, in order to classify the tame representation type hereditary skew group algebras, it suffices to study the group actions on the euclidean quivers. It is necessary to clarify that the case will be considered separately later on.

Before we present the results, we need some definitions.

Definition 4.1.We say that an quiver of type() has

symmetric orientation ifis odd and the quiver issymmetric with respect to an axis,

cyclic orientation of orderif the full subquivers with verticesare all equal, andis minimal with respect to this property().

Remark 4.2.Supose you havewith a fixed oritentation. Choosesuch that, for any action. This set, a non-empty setof the natural numbers, has a first element and this is theof thedefinition.

Definition 4.3.We say that an quiver of type,, has

symmetric orientation of kindif

, or

, or

, or

,

symmetric orientation of kindifis even and thequiver is symmetric with respect to the middle point;

Definition 4.4.We say that an quiver of typehas symmetricorientation of orderif the number of arrows starting at the vertexis equal to, for.

Definition 4.5.We say that an quiver of typehas

symmetric orientation of kindif,,or,andis a source or a sink;

symmetric orientation of kindif it is not symmetric ofkind (a) and it is symmetric with respect to the side.

Definition 4.6.We say that an quiver of typehas symmetricorientation if it is symmetric with respect to the side.

Remark 4.7.

We say that the quiveris symmetric with respect to themiddle pointif that point is center of symmetry of thequiver.

We say that the quiveris symmetric with respect to an axis, if the line obtained with the pointsissymmetry axis of the quiver.

Let be a group and we will assume that is acting trivially on , we have . Hence, from now on, we will assume that is acting non trivially on . Let . Clearly is a normal subgroup of . Let , then is a short exact sequence of groups.

has symmetric orientation of kind (a), the order ofis divisible byorandor;

or

has symmetric orientation of kind (b), the order ofis divisible byand;

Ifandwithof typethenhassymmetric orientation,is a group of even order and;

Ifwithof typethenand.

Proof. In order to prove the theorem, we need a precise description of all the possible actions of on , for each type and orientation of . We use Proposition 2.1 to describe all possible actions of on with of type , , , or . We observe that we identify the elements of with the natural numbers in the indexes of the idempotents .

Theorem 2.8 cannot be applied because is not simply connected. Using [16, (2.3), (2.4)] for this case we have (where , , , under the conditions of [16, (2.3)]) .

Let with of type and let , . Assume that fixes at least one point, say . Then or . In the first case, repeating this procedure we have that for all , and so , a contradiction. In the second case, we get that , for all , and this determines the orientation of the arrows. If , we have and , a contradiction since there is only one arrow joining and . So and has symmetric orientation. Moreover, for all , so .

Now let , , for all . Let . If , the previous reasoning says that there must exist a middle point between and that will be fixed by , a contradiction. So , and inductively we get that . This determines the orientation of the arrows, and so is cyclic of order , where is the first element in the set . Let be such that . Let , with . Then . If , we get a contradiction to the minimality of . So and and in .

We denote by

We have already proved that if and only if has symmetric orientation, and if and only if is cyclic of order .

Assume first that is cyclic of order and is not symmetric. We have seen that in for any . Moreover, , so if and only if is divisible by . Let be the smallest natural number such that is divisible by . We conclude that in this case.

Assume now that is symmetric but not cyclic, and let , that is, and for some and , . We assume, without loss of generality, that . If , then , so is divisible by . Since , we have that for . If then and hence , that is, in , and hence in this case. If then and is symmetric with respect to the axes and and in this case .

Finally, assume that is symmetric () and cyclic of order , and let and , that is, and . If , then , so is divisible by . Since , we have that and in this case and .

Finally, from Theorem 2.2 and [16, (2.3)] we have the conclusions, that is or .

Let with of type , . Let , . We observe first that , see Proposition 2.1.

Assume first that and . Then for all . Since , we must have or . This implies that has symmetric orientation of kind and all possible actions are given by:

, , , ;

, , , ;

, , , .

Since and , we conclude that or .

Assume now that and . Then, using the same argument as in the proof of Theorem 3.6 in the case of , we conclude that is symmetric of kind . If is not of kind , the unique possible non trivial action on the complete set of idempotents is given by , , , and for all . In this case, .

To finish with this case, we have to assume that is symmetric of kind and . Then all the possible non trivial actions are given by

, , , , for all ;

, , , , for all ;

, , , , for all ;

, , , , for all ;

, , , , for all ;

, , , , for all ;

, , , , for all .

Now , , and . Moreover, if , then implies that is equal to , , , , or . Moreover . Hence , or .

This case follows from an argument similar to what has been done in the proof of Theorem 3.6 for the case (for any , and the action of on is uniquely determined by the action of in and ).

Let with of type , and let . By Proposition 2.1, and then . Now or . In the first case we get that for all , and so , a contradiction. Then and this determines completely the orientation of the arrows, that is, has symmetric orientation, and the action of on the complete set of idempotents of . Since , we can deduce that is an even number. Let , . By the previous reasoning, and act in the unique possible way on the complete set of idempotents of . Then for all , hence , that is, in . So and , see Theorem 2.2 and Theorem 2.8.

If we consider the case of type , the unique possible action on the set of idempotents is the trivial one. Hence , and the result follows from i).

The case is not considered in Theorem 4.8 because the techniques we use do not hold in this case. In fact, is the Kronecker algebra and Proposition 2.1 does not hold since this algebra has double arrows. Moreover, Theorem 2.8 cannot be applied because is not simply connected. We will only consider the case of a cyclic group acting on the Kronecker algebra, then it is possible to apply directly [16, (2.3)].

If is a cyclic group acting on , the Kronecker algebra, with invertible in then all possible actions are given by: